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I'm having some difficulties understanding the Langevin thermostat (MD).

In my notes, there is written that the Langevin equation is

$$ m\dot{v} = F - m\gamma v + f_R, \tag{1} \label{1}$$

where $f_R$ is the random force. Then, to prove that by using this thermostat we are actually sampling from the canonical ensemble, there is written that the starting point of the proof are the overdamped Langevin equations:

$$ \dot{r}_i(t) = \beta D F_i (r_1(t), \dots,r_N(t)) + \eta_i(t), \tag{2} \label{2}$$

where $\eta_i(t)$ is a random velocity. Where do $\eqref{2}$ come from? Are they derived from $\eqref{1}$?

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Eqn (2) basically comes from setting $m\dot{v}=0$ in eqn (1), on the assumption that the velocities are strongly damped. Then the $m\gamma v$ term is taken over to the left side, and both sides are divided by $m\gamma$. The diffusion coefficient is $D=k_BT/m\gamma$.

Having said all that, this is not a proof that the thermostat generates the canonical ensemble. Such a proof usually (I believe) starts by considering the Fokker-Planck equation (in both $r$ and $v$) which is equivalent to the Langevin equation, and showing that the canonical ensemble (with a potential from which $F$ is derived) is a stationary solution of that equation.

Eqn (2) can lead to a proof that the positions are sampled from the canonical ensemble, in this overdamped limit, but to show that both positions and velocities are correctly sampled in general, one should keep the velocity evolution of eqn (1).

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